Question

If a connected 3-regular planar graph *G* has 18
vertices, then the number of faces of a planar representation of
*G* is..........

Answer #1

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Suppose that a connected planar graph has eight vertices each of
degree 3 then how many regions does it have?And suppose that a
polyhedron has 12 triangular faces then determine the number of
edges and vertices.

Let there be planar graph G with 12 vertices where every
vertices may or may not be connected by an edge. The edges in G
cannot intersect. What is the maximum number of edges in G. Draw an
example of G. What do you notice about the faces and the maximum
number of edges?

Suppose we are going to color the vertices of a connected planar
simple graph such that no two adjacent vertices are with the same
color.
(a) Prove that if G is a connected planar simple graph, then G
has a vertex of degree at most five.
(b) Prove that every connected planar simple graph can be
colored using six or fewer colors.

G is a complete bipartite graph on 7 vertices.
G is planar, and it has an Eulerian path. Answer the questions, and
explain your answers.
1. How many edges does G have?
2. How many faces does G have?
3. What is the chromatic number of G?

Let G be a connected planar graph with 3 or more vertices which
is drawn in the plane. Let ν, ε, and f be as usual. a) Use P i fi =
2ε to show that f ≤ 2ε 3 . b) Prove that ε ≤ 3ν − 6. c) Use b) to
show that K5 is not planar.

30. a) Show if G is a connected planar simple graph with v
vertices and e edges with v ≥ 3 then e ≤ 3v−6.
b) Further show if G has no circuits of length 3 then e ≤
2v−4.

show that any simple, connected graph with 31 edges and 12 vertices
is not planar.

Use proof by induction to prove that every connected planar
graph with less than 12 vertices has a vertex of degree at most
4.

Prove that every connected planar graph with less than 12
vertices can be 4-colored

Prove or disprove the following:
(a) Every 3-regular planar graph has a
3-coloring.
(b) If ?=(?,?) is a 3-regular graph and there
exists a perfect matching of ?, then there exists a set of edges
A⊆E such that each component of G′=(V,A) is a cycle

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